Logic puzzle: Child gender compounded with irrelevant data

Select all people who have two children, at least one of which is a boy born on Tuesday. Given one of those individuals, what is the probability that their other child is a boy?

Common sense suggests there are only two possible outcomes: TuesdayBoy+Boy, and TuesdayBoy+girl, each with a probability of 1/2.

But some hard-charging pundit for the other team has gone on to suggest otherwise by creating an outcome space like so:

Mon-Boy

Tue-Boy

Wed-Boy

Thu-Boy

Fri-Boy

Sat-Boy

Sun-Boy

Mon-Girl

Tue-Girl

Wed-Girl

Thu-Girl

Fri-Girl

Sat-Girl

Sun-Girl

Mon-Boy

XXXXXX

Tue-Boy

XXXXXX

XXXXXX

XXXXXX

XXXXXX

XXXXXX

XXXXXX

XXXXXX

XXXXXX

XXXXXX

XXXXXX

XXXXXX

XXXXXX

XXXXXX

XXXXXX

Wed-Boy

XXXXXX

Thu-Boy

XXXXXX

Fri-Boy

XXXXXX

Sat-Boy

XXXXXX

Sun-Boy

XXXXXX

Mon-Girl

XXXXXX

Tue-Girl

XXXXXX

Wed-Girl

XXXXXX

Thu-Girl

XXXXXX

Fri-Girl

XXXXXX

Sat-Girl

XXXXXX

Sun-Girl

XXXXXX

From this table of outcomes, 27/196 possible two-child families have at least one Tuesday-Boy. Furthermore, 13/196 outcomes are TuesdayBoy+Boy, while 14/196 outcomes are TuesdayBoy+girl. And I'm willing to believe that drawing a random entry from this outcome space will result in TuesdayBoy+Boy with a probability of 13/196, and TuesdayBoy+Girl with a probability of 14/196.

But I'm pretty sure the assessment that this represents the quoted question accurately is in error. And I think the mechanism of selecting the subset of qualified results is the source of that error.

Re: Logic puzzle: Child gender compounded with irrelevant data

I will agree that 13/196 possibilities in the example outcomes space are boy-boy.

But this seems like a misuse of a mathematical construct to me. Substitute any irrelevant fact into the table, and we get an equally arbitrary answer:

I flipped a fair coin twice, and I can tell you about one of the results: One outcome was a head; also, I didn't own any cats at the time. What are the odds the other result was also a head? [P(OwnCat) declared as 1/2 my lifetime]